AI 中文总结
本文通过矩方法刻画了强正则图上的均匀平均混合，给出了存在性判据、显式时间分布，并修正了瞬时均匀混合的分类。
AI 中文摘要
我们研究了在非完全连通强正则图上的连续时间量子游走，观测时间从自由选择的概率分布中随机抽取。均匀平均混合（UAM）要求存在一个概率分布，使得所有平均转移概率都等于 $1/n$，其中 $n$ 是顶点数。在强正则图上，这等价于三个余弦矩上的两个仿射约束。我们为每个具有非整数特征值的强正则图构造了一个有界、紧支撑的时间密度。对于整数谱，我们给出了一个精确的有限Toeplitz准则及其Hankel形式。此类图的每个平均混合矩阵最多由两个观测时间实现。矩线上的三个基本不等式（同时也给出了Chan在Bose-Mesner代数中复Hadamard矩阵分类的简短证明）导致了所有允许UAM的强正则图的确定。除了非平方阶的会议图以及具有瞬时均匀混合的图之外，这些图是两个无限参数族及其补图的成员，并且我们为它们给出了具有两个观测时间的显式分布。Petersen图及其补图是最小的成员。具有UAM的强正则图恰好在其没有瞬时均匀混合时允许有界时间密度。我们还修正了Godsil、Mullin和Roy对强正则图上瞬时均匀混合的分类。其符号条件排除了半5-立方体，该图在时间 $\pi/4$ 时均匀混合。若按字面意义读取阶数 $4\theta^2$，其奇偶条件也排除了Clebsch图，并包含了参数 $(36,14,4,6)$，对于该参数集，没有任何时间分布能给出均匀平均混合。
英文摘要
We study continuous-time quantum walks on connected strongly regular graphs that are not complete, observed at a random time drawn from a freely chosen probability law. Uniform average mixing (UAM) asks for a law under which every averaged transition probability equals $1/n$, where $n$ is the number of vertices. On a strongly regular graph this is equivalent to two affine constraints on three cosine moments. We construct a bounded, compactly supported time density for every strongly regular graph with nonintegral eigenvalues. For integral spectra we give an exact finite Toeplitz criterion and its Hankel form. Every averaged mixing matrix of such a graph is realized by at most two observation times. Three elementary inequalities on the moment line, which also give a short proof of Chan's classification of complex Hadamard matrices in the Bose-Mesner algebra, lead to a determination of all strongly regular graphs that admit UAM. Apart from the conference graphs of nonsquare order and the graphs with instantaneous uniform mixing, these are the members of two infinite families of parameter sets and their complements, and for them we give explicit laws with two observation times. The Petersen graph and its complement are the smallest members. A strongly regular graph with UAM admits a bounded time density exactly when it has no instantaneous uniform mixing. We also correct the classification of instantaneous uniform mixing on strongly regular graphs by Godsil, Mullin and Roy. Its sign condition excludes the halved $5$-cube, which mixes uniformly at time $π/4$. With the order $4θ^2$ read literally, its parity condition also excludes the Clebsch graph and includes the parameters $(36,14,4,6)$, for which no time law gives uniform average mixing.
Comments26 pages, 1 table. Includes a correction of Theorem 5.1 and Lemma 5.2 in arXiv:1301.5889
